Eigenvalues for Infinite Matrices, Their Computations and ... · matrices and Fredholm integral equations),John von Neumann (1929, In nite matrices as operators) andRichard Bellman(1947,

Eigenvalues for Infinite Matrices, TheirComputations and Applications

P.N.Shivakumar

Email: [email protected]

(Collaborator: Yang Zhang)

Department of Mathematics

University of Manitoba,Winnipeg, Canada R3T 2N2

BIRS Workshop March 22-27, 2015 P.N.Shivakumar 1

Abstract

In this talk, we discuss

• properties of eigenvalues which occur in infinite linear algebraic

systems of the form Ax = λx , where A is an infinite diagonally

dominant matrix and b is a bounded linear operator,

• the classical question “Can you hear the shape of a drum?”:

“YES” for certain convex planar regions with analytic

boundaries; “NO” for some polygons with reentrant corners,

• eigenvalues and van der Waerden infinite matrix.


History

Earliest concept of infinity comes from Anaximander, a Greek and

consequently mathematical infinity is attributed to ZENO (400

B.C.). Also. The Indian text Surya Prajnapti (300-400 B.C.)

classified all numbers into three sets, namely, enumerable,

innumerable, and infinite. In 1655, Waller introduced the symbol

∞.

“Two things are infinite: the universe and human stupidity; I am

not sure about the universe.” ——– A. Einstein


Discussion of infinite systems generally start with truncated finite

systems.

References:

(a) “A History of Infinite Matrices” by Michael Bernkopf

(b) “Infinite Matrices and Sequence Space” by R. G. Cooke


Some earlier authors are Poincare (1984, Hill’s equation), Hilbert

(1906, Hilbert infinite quadratic forms, equivalent to infinite

matrices and Fredholm integral equations), John von Neumann

(1929, Infinite matrices as operators) and Richard Bellman (1947,

The boundedness of solutions of infinite systems of linear

equations).


We give three different criteria for An×n = (aij) to be nonsingular.

(a) Diagonal dominance

σi |aii | =n∑

j=1, j 6=i

|aij |, 0 ≤ σi < 1, i = 1, 2, ..., n.

(b) A chain condition

For A = (aij)n×n, there exists, for each j /∈ J, a sequence of

nonzero elements of the form ai ,i1 , ai ,i2 , ..., ai ,ij with j ∈ J.

(c) Some sign patternsaij > 0 for i ≤ j ,

(−1)i+jaij > 0 for i > j ,(i , j = 1, 2, ..., n)


Linear Algebraic Systems

Consider

∞∑j=1

aijxj = bi , i = 1, 2, ...,∞,

where

• the infinite matrix A = (aij),

• σi |aii | =∑∞

j=1 |aij |; 0 ≤ σi < 1, i = 1, 2, ...,∞,

• the sequence bi is bounded.

We establish sufficient conditions which guarantee the existence

and uniqueness of a bounded solution to the above system as well

as bounds for truncated systems.


We have

|x (n+1)j − x

(n)j | ≤ Pσn+1 + Q|an+1,n+1|−1

for some positive constraints P and Q.

An estimate for the solution is given by

|x (n)| ≤n∏

k=1

1 + σk1− σk

n∑k=1

|bk ||akk |(1 + σk)

.


Linear Eigenvalue Problem

To determine the location of eigenvalues for diagonally dominant

infinite matrices (which occur in (a) solutions of elliptic partial

differential equations and (b) solutions of second order linear

differential equations) and establishes upper and lower bounds of

eigenvalues, we define an eigenvalue of A to be any scalar λ for

which Ax = λx for some 0 6= x ∈ D(A).


Assume A is an operator on l1.

E (1): aii 6= 0, i ∈ N and |aii | → ∞ as i →∞.

E (2): ∃ρ ∈ [0, 1), s.t. for each j ∈ N

Qj =∞∑

i=1,i 6=j

|aij | = ρj |ajj |, 0 ≤ ρj ≤ ρ.

E (3): For all i , j ∈ N, i 6= j , |aii − ajj | ≥ Qi + Qj .

E (4): For all i ∈ N, sup|aij | : j ∈ N <∞.


Linear Differential System

We consider the infinite system

d

dtxi (t) =

∞∑j=1

aijxj(t) + fi (t), t ≥ 0, xi (0) = yi , i = 1, 2, ...

In particular, if A is a bounded operator on l1, then the

convergence of a truncated system has been established.

(Arley and Brochsenius (1944), Bellman (1947) and Shaw (1973)

have studied this problem. However, none of these papers yields

explicit error bounds for such a truncation. )


Applications

1. Digital circuit dynamics

2. Conformal mapping of doubly connected regions

3. Fluid flow in pipes

4. Mathieu equation d2ydx2 + (λ− 2q cos 2x)y = 0 for a given q

with the boundary conditions: y(0) = y(π2 ) = 0

(Our interest: two consecutive eigenvalues merge and become

equal for some values of the parameter q. )


5. Bessel Function

6. Eigenvalues of the Laplacian on an elliptic domain

∂2u

∂x2+∂2u

∂y2+ λ2u = 0 in D, u = 0 on ∂D,

where D is a plane region bounded by smooth curve ∂D.

Consider the domain bounded by the ellipse represented byx2

α2 + y2

β2 = 1, which can be expressed correspondingly in the

complex plane by

(z + z)2 = a + bzz ,

where a = 4α2β2

β2−α2 , b = 4α2

α2−β2 .


After considerable manipulation, we get the value of u on the

ellipse as

u = 2a0 +∞∑n=1

A2n,0B0,nan +∞∑k=1

(−λ

2

4

)k2a0

k!k!(zz)k

+∞∑k=1

[k∑

n=1

(A2n,kb0,n +

n∑l=1

A2n,k−lbl ,n

)an

](zz)k

+∞∑k=1

[ ∞∑n=k+1

(A2n,kb0,n +

k∑l=1

A2n,k−lbl ,n

)an

](zz)k .


For u = 0 on the elliptic boundary, we equate the powers of zz to

zero, where we arrive at an infinite system of linear equations of

the form

∞∑k=0

dknan = 0, n = 0, 1, 2, ...,∞,

where dkn’s are known polynomials of λ2. The infinite system is

truncated to n × n system and numerical values were calculated

and compared to existing results in literature. The method

derived here provides a procedure to numerically calculate the

eigenvalues.

8. Mathieu Equation


Shape of A Drum

For the classical question, “Can you hear the shape of a drum?”,

the answer is known to be “yes” for certain convex planar regions

with analytic boundaries. The answer is also known to be “no” for

some polygons with reentrant corners. A large number of

mathematicians over four decades have contributed to the topic

from various approaches, theoretical and numerical.


We develop a constructive analytic approach to indicate how a

preknowledge of the eigenvalues leads to the determination of the

parameters of the boundary. This approach is applied to a general

boundary and in particular to a circle, an ellipse, an annulus and a

square. In the case of a square, we obtain an insight into why the

analytical procedure does not, as expected, yield an answer.


The pursuit of a “complete” solution to the question “Can one

hear the shape of a drum”, originally posed by Lipman Bers and

used as a title in 1966 by Mark Kac, has been a fascinating journey

and work can only be described as “work in progress”. The

research has involved many mathematicians and many tools

including Asymptotics, Probability Theory, Operator Theory,

infinite algebraic systems and inevitably intense computational

work involving approximate methods.


Mathematically, the problem is, whether a preknowledge of the

eigenvalues of the Laplacian in a region Ω leads to the definition of

Γ, the closed boundary of Ω. Specifically, we have

uxx + uyy + λ2u = 0 in Ω, (1)

u = 0 on Γ. (2)

According to the maximum principle for linear elliptic partial

differential equations, the infinite eigenvalues λ2n, n = 1, 2, 3, ...,

are positive, real, ordered and satisfy

0 < λ21 < λ2

2 < · · · < λ2n < · · · <∞. (3)


In the famous paper “Can you hear the shape of a drum?”, Mark

Kac makes an analysis using only asymptotic properties of large

eigenvalues and uses probability theory as the tool to establish that

one can hear the area of a polygonal drum and conjectures for

multiply connected regions.

A major contribution in 2000 is given by Zelditch, where a positive

answer “yes” is given for certain regions with analytic boundaries.


Preliminaries

We express (1) in terms of complex variables z = x + iy ,

z = x − iy , and the system (1) and (2) becomes

uzz +λ2

4u = 0 in Ω, (4)

u = 0 on Γ. (5)


We start the analysis using the completely integrated form of thesolution to (1) as given in Vekua by

u =

f0(z)−

∫ z

0f0(t)

∂

∂tJ0

(λ√

z(z − t))dt

+ conjugate, (6)

where f0(z) is an arbitrary analytic function which can be formallyexpressed as

f0(z) =∞∑n=0

anzn (7)

and J0 represents the Bessel function of first kind and order 0given by

J0

(λ√

z(z − t))

=∞∑q=0

(−λ

2

4

)zq(z − t)q

q!q!.


From an identity given in Abramowitz and Stegun, we have, when

n is even,

zn + zn =

n2∑

m=0

cmn(z + z)n−2m(zz)m, n = 2, 4, 6, ...,


Boundary Γ

(a) A general boundaryWe consider the parametrized analytical boundary Γ with biaxialsymmetry to be given by

(z + z)2n =∞∑n=0

dn1(zz)n

which yields, on using Cauchy products for infinite series,

(z + z)2np =∞∑n=0

dnp(zz)n,

where

dnp =n∑

l=0

dl p−1dn−l 1, p = 1, 2, 3, ....


(b) Circular boundary

We consider the circular boundary given by

x2 + y2 = a2 or zz = a2.

(c) An elliptic boundary

We consider the elliptic boundary Γ given by

x2

α2+

y2

β2= 1 or (z + z)2 = a + bzz , α > 0.


(d) A square boundary

We give this example to demonstrate why our analytical approach

does not yield information of the boundary with sharp corners from

a preknowledge of eigenvalues. Consider a square boundary given

by

x = ±a, y = ±a or z4 + z4 = 2(zz)2 − 16a2(zz) + 16a4

or z2 + z2 = 4(zz − 2a)2.


We assume

f0(z) =∞∑n=0

a2nz2n,

u = 2a0J0

(λ√zz)

+∞∑n=1

a2n

∞∑k=0

(−λ

2

4

)A2n k(z2n + z2n)(zz)k ,

which on substitution gives

u = 2a0J0(λ√zz)+

∞∑n=1

a2n

∞∑k=0

(−λ

2

4

)A2n k

n∑m=0

(−1)m2n(2n −m − 1)!

m!(2n − 2m)!(z + z)2(n−m)(zz)m.


(a) General boundary Γ

After rearrangement of summations, we get

u = 2a0

+∞∑n=1

a2nDn 0 0 0 +

[2a0

(−λ

2

4

)+∞∑n=1

a2n

1∑

i=0

1−i∑p=0

Dn p 1−i−p i

]zz

+∞∑q=2

2a0

(−λ

2

4

)q1

q!q!+

q−1∑n=1

a2n

[n∑

i=0

q−1∑p=0

Dn p q−i−p i

]

+∞∑

a2n

[q∑ q−1∑

Dn p q−i−p i

](zz)q.


Now we equate the coefficients of (zz)q, q = 0, 1, 2, ... to zero to

obtain the infinite linear algebraic system. Writing the resulting

system as

∞∑n=0

fqna2n, q = 0, 1, 2, ...,

we note that fqn is a polynomial of degree q in µ and contains only

d01, d11, d21,.... The values of µ are determined by formally

setting D = det(fqn) = 0, which can be expressed as an infinite

series in µ given by

D =∞∑i=0

Diµi

and µi ’s are given by D = 0.


We note that we can write

D = D0

∞∏i=1

(1− µ

µi

)

formally suggesting that DiD0

is the sum of the inverse products of

µ1, µ2,... taken i at a time. In particular,

D1

D0= −

∞∑i=1

1

µi,

D2

D0=∞∑i=1

∞∑j=i+1

1

µiµj


(b) A circle

It is well known that the solutions to (1) and (2) is given by

u = a0J0

(λ√zz)

and the eigenvalues are given by the zeros of J0(λa) = 0, namely,

λi =j0ia, i = 1, 2, ....

We can write

∞∑q=0

(−λ

2

4

)q1

q!q!a2q =

∞∏j=1

(1− λ2

λ2j

).


We obtain

a2 = 4∞∑j=1

1

λ2j

,

thus establishing uniquely the value of a if all the eigenvalues λ1,

λ2,... are known. In fact, substitute for λi from λi = j0ia we obtain

a2

4=∞∑j=1

a2

j20 j

which checks with the well-known fact that

∞∑j=1

1

j20 j

= 4.


(c) An ellipse

Here we develop the solution in the elliptic domain. We get

∞∑n=0

fnqa2n = 0, q = 0, 1, 2, ....

We note that fnq is a polynomial of degree q in µ. The

determinant D1 is the sum of an infinite determinants derived from

differentiating D with respect to µ and setting µ = 0.


(d) A square boundary

We give this example to demonstrate how our approach does notyield information of the boundary from a preknowledge of theeigenvalues in the case of a boundary containing sharp corners.Consider the square boundary

z2 + z2 = 4(zz − 2a)2.

We use f0(z) =∑∞

n=0 a4nz4n to obtain

u = 2a0J0

(λ√zz)

+∞∑n=1

∞∑k=0

(−λ

2

4

)k

A4n k(z4n + z4n)a4n(zz)k ,

z4n + z4n =n∑

m=0

bmn(z2 + z2)2n−2m(zz)2m,

which gives z4n + z4n =∑n

m=0 bmn4n−m(zz − 2a)n−m.


It is well-known that for a square, the eigenvalues λ2 are given bym2π2

a2 + n2π2

a2 and using µ = −λ2

4 yields

D1

D0= −

∞∑i=1

1

µi=

4a2

π2

∞∑m=1

∞∑n=m+1

1

m2 + n2.

The double series is clearly divergent and hence the failure of the

analytic approach, although det[gqn] = 0 yields the eigenvalues.


Conclusions

In this article, we have demonstrated a constructive approach when

the answer is “yes”. We give an interpretation to “a preknowledge

of eigenvalues” as knowing the sums of the inverse products of the

eigenvalues µi , i = 1, 2, ..., taken i at a time. In the case of an

ellipse, we express the parameters of the ellipse in terms of the

eigenvalues. Our conjecture is that for the answer to be “yes”, it is

necessary for the sums of the inverse products of the eigenvalues

µi , i = 1, 2, ..., taken i at a time to be convergent series. Another

conjecture is that all analytic curves (curvilinear polygons for

example) yield the answer “yes”.


Hearing the Shape of a Drum with a Hole

We are interested in the Helmholtz equation

d2u

dx2+

d2u

dy2+ λu = 0 in Ω

where Ω is the doubly connected region with the boundary

conditions

u = 0 on π1 : x2 + y2 = a2

u = 0 on π2 : x2 + y2 = b2 a > b


We need to show that if the eigenvalues of λ are known, the ratio

of the annulus is uniquely determined.

Equation d2udx2 + d2u

dy2 + λu = 0 in polar coordinates (r , θ) and noting

that u = u(r), we get the Bessel equation

d2u

du2+

du

dr+ λu = 0

whose solution is given in Ω [Garabedian].


U = c1J0(√λr) + c2

J0(√λr) ln(

√λr)−

∞∑n=1

anHnλnr2n

where an = (−1)n

2nn! and Hn = 1 + 12 + 1

3 + · · ·+ 1n .

Using the previous equations, we get

c1J0(√λa) + c2

J0(√λa) ln(

√λa)−

∞∑n=1

anHnλna2n

= 0

c1J0(√λb) + c2

J0(√λb) ln(

√λb)−

∞∑n=1

anHnλnb2n

= 0


For non-zero c1 and c2, we need

J0(√λa)J0(

√λb) ln

(b

a

)+

J0(√λb)

∞∑n=1

anHnλna2n − J0(

√λa)

∞∑n=1

anHnλnb2n

= 0

Equation above becomes an infinite series in λ. In fact, we can

express the infinite series in the above equation by

∞∏j=1

(1− λ

λj

)ln

(b

a

)


Equating the constants, we get ln(ba

)= ln

(ba

). And equating the

coefficient of λ, we get

a1(a2 + b2) ln

(b

a

)+ a1H1(a2 − b2) = −

∞∑j=1

1

λjln

(b

a

).

Using H1 = 1, a = 1 and a1 = − 124 , we get

1

24(1 + b2) +

1− b2

ln(b)=∑ 1

λj= P.


Knowing P, we can numerically solve for b. We also note that

conversely, if b is known, we get the value of

∞∑j=1

1

λj.

Hence, we can hear the shape of the drum which is an annulus if

the eigenvalues are known.


Zeros of Infinite Series and Ax = b

For the infinite series in λ, we will assume that the zeros of the

series λn, n = 1, 2, 3, ...; we can write, with λ2 replaced by x as

y(x) =∞∑n=0

cnx2n =

∞∏n=1

(1 + anx)

where c0 = 1, c1 =∞∑k=1

aky′(0) and λn = − 1

an.

We also have 0 < λ21 < λ2

2 < λ23 < · · · . We seek to connect a’s

with cn’s.


The solution to

y ′′ + f (x)y = 0, (8)

when f (x) is an arbitrary function with continuous derivatives, is

expressed by

∞∑k=0

cnxn, (9)

where cn’s are explicitly dependent on derivatives of f (x) and the

initial conditions on y(x), namely, y(0) and y ′(0).

In fact, y(0) = c0 and y ′(0) = c1.

Reference: P. N. Shivakumar and Y. Zhang, On series solutions of y ′′–f (x)y = 0 and

applications, Advances in difference equations, 2013, 201: 47.


It can be easily shown that, with

f (x) =

( ∞∑n=1

an1 + anx

)2

−∞∑n=1

a2n

(1 + anx)2, (10)

solution to (8) is given by (9) in terms of Qp.

We will use the notation Qp =∞∑k=1

apk , p = 1, 2, 3, ...


Finally, we get

∞∑k=1

apk = ep, p = 1, 2, ..., (11)

where ep’s are explicitly obtained as combinations of cn’s.

Rewriting the above equations as

∞∑k=1

apkxk = ep, p = 1, 2, ...,

we note that the system is of the form Ax = e, where x is a unite

vector and A is the well-known Vandermonde infinite matrix. We

now get x = A−1e. In all the discussion above, infinity can be

replaced by a positive integer.


As a simple example, we consider a polynomial of degree 3 with

increasing positive roots. By the above analysis, we arrive at the

system

x1 + x2 + x3 = c1,

x21 + x2

2 + x23 = c2,

x31 + x3

2 + x33 = c3,

where c1, c2 and c3 are determined by the coefficients of the

polynomial. We arrive at the bound

0 < x1 <c2

c1< x2 <

c3

c2< x3 <

c1

3.


References

• P. N. Shivakumar and Y. Wu, Eigenvalues of the Laplacian on

an elliptic domain, Comput. Math. Appl. 55(2008)1129-1136.

• P. N. Shivakumar, Diagonally dominant infinite matrices in

linear equations, Utilitas Mathematica, Vol. 1, 1972, 235-248.

• P. N. Shivakumar and R. Wong, Linear equations in infinite

matrices, Journal of Linear Algebra and its Applications, Vol. 7,

1973, No. l, 553-562.

• P. N. Shivakumar and K. H. Chew, A sufficient condition for the

vanishing of determinants, Proceedings of the American

Mathematical Society, Vol. 43, No. 1, 1974, 63-66.


• P. N. Shivakumar, K. H. Chew and J. J. Williams, Error bounds

for the truncation of infinite linear differential systems, Journal

of the Institute of Mathematics and its Applications, Vol. 25,

1980, 37-51.

• P. N. Shivakumar, N. Rudraiah and J. J. Williams, Eigenvalues

for infinite matrices, Journal of Linear Algebra and its

Applications,1987 Vol. 96, 35-63.

• P. N. Shivakumar and J. J. Williams, An iterative method with

truncation for infinite linear systems, Journal of Computational

and Applied Mathematics, 24, 1988, 109-207.

• P. N. Shivakumar and C. Ji, On Poisson’s equation for doubly

connected regions , Canadian Applied Mathematics Quarterly,

Vol.1, No. 4, Fall 1993


• P. N. Shivakumar and C. Ji, Upper and lower bounds for inverse

elements of finite and infinite tri-diagonal matrices, Linear

Algebra and its Applications 247:297-316, 1996

• P. N. Shivakumar and Q. Ye, Bounds for the width of instability

intervals in the Mathieu equation, accepted for publication in

the Proceedings of the International Conference on Applications

of Operator Theory, Birkhauser series “Operator Theory:

Advances and Applications”, Vol.87, 348- 357, 1996.

• P. N. Shivakumar, J. J. Williams, Q. Ye and C. Marinov, On

two-sided bounds related to weakly diagonally dominant

M-matrices with application to digital circuit dynamics, accepted

for publication in Siam J. Matrix Analysis and Applications,

Vol.17, No.2, 298-312, April 1996.


• P. N. Shivakumar and C. Ji, On the nonsingularity of matrices

with certain sign patterns, Linear Algebra and its Applications

273:29-43, 1998.

• P. N. Shivakumar and A. G. Ramm, Inequalities for the minimal

eigenvalue of the Laplacian in an annulus, Mathematical

Inequalities & Applications, Vol.1, No. 4, 559-563. 1998.

• P. N. Shivakumar and J. Xue, On the double points of a

Mathieu equation, Journal of Computational and Applied

Mathematics, 109, 111-125, 1999.


• P. N. Shivakumar, Y. Wu and Y. Zhang, Shape of a drum, a

constructive approach, Transactions on Mathematics, WSEA,

2011.

• P. N. Shivakumar and Y. Zhang, On series solutions of

y ′′–f (x)y = 0 and applications, Advances in difference

equations, 2013, 201: 47.

• P. N. Shivakumar and K. C. Sivakumar, A review of infinite

matrices and their applications, Linear Algebra and its

Applications 430 (2009) 976-998.


Thank you!


Documents

Eigenvalues for Infinite Matrices, Their Computations and ... · matrices and Fredholm integral equations),John von Neumann (1929, In nite matrices as operators) andRichard Bellman(1947,